ath eth Oper Res (2016) 83:179 205 DOI 10.1007/s00186-015-0525-x ORIGINAL ARTICLE odelng values for TU-games usng generalzed versons of consstency, standardness and the null player property Tadeusz Radzk 1 Theo Dressen 2 Receved: 23 Aprl 2013 / Accepted: 30 November 2015 / Publshed onlne: 23 December 2015 The Author(s) 2015. Ths artcle s publshed wth open access at Sprngerlnk.com Abstract In the paper we dscuss three general propertes of values of TU-games: λ-standardness, general probablstc consstency and some modfcatons of the null player property. Necessary and suffcent condtons for dfferent famles of effcent, lnear and symmetrc values are gven n terms of these propertes. It s shown that the results obtaned can be used to get new axomatzatons of several classcal values of TU-games. Keywords TU-game Null player axom λ-standardness Consstency Probablstc consstency The Shapley value The per-capta value The soldarty value The equal splt value 1 Introducton Classcal cooperatve game n the characterstc functon form (also called a TU-game) s a functon v : 2 N R wth a fnte set N as the grand coalton of the players (agents, ndvduals). For each coalton S (a subset of N), v(s) represents the worth Ths paper was supported by GRANT S30103/I-18 of Wrocław Unversty of Technology. B Tadeusz Radzk tadeusz.radzk@pwr.edu.pl Theo Dressen t.s.h.dressen@ew.utwente.nl 1 Insttute of athematcs and Computer Scence, Wrocław Unversty of Technology, Wybrzeże Wyspańskego 27, 50-370 Wrocław, Poland 2 Department of Appled athematcs, Faculty of Electrcal Engneerng, athematcs and Computer Scence, Unversty of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
180 T. Radzk, T. Dressen of S (the gan possble to be acheved jontly by all the players from S when they collaborate). One of the essental problems addressed n the papers on ths topc s the followng: Assumng that all the players n N wll collaborate and create (fnally) the grand coalton N, how to dvde farly the whole worth v(n) between them? In the lterature, many varous procedures (called values) have been proposed for solvng ths problem, and the classcal Shapley value (Shapley 1953) s the most promnent and the best known one among them. However ths value has the so-called null player property that does not allow for soldarty between the players, assgnng zero payoff to every unproductve player ; that s to every player wth all hs margnal contrbutons v(s ) v(s) = 0forS N (called then a null player). On the opposte sde, there s a very classcal value allowng for the total soldarty between the players, whch radcally rejects the null player property the so-called equal splt value. Itawards every player n the grand coalton N, ndependently of hs contrbutons, v(s ) v(s) to coaltons S, the same payoff equal to v(n) n where n s the cardnalty of N. Inthe lterature, many other values for TU-games standng between the Shapley value and the equal splt value were constructed and characterzed by proper sets of axoms. The most nterestng results here are axomatzatons of the class of convex combnatons of those two values (called egaltaran Shapley values) found n the two recent papers by van den Brnk et al. (2013) and Casajus and Huettner (2013). Two of the propertes used most frequently to axomatze the values of cooperatve games are the null player axom and consstency. The consstency property for a value states that f all the players are supposed to be pad accordng to a payoff vector n the orgnal game, then the players n every reduced game (wth a smaller number of players) can acheve some payoff vector closely related wth the value of the reduced game. A survey of consstency propertes of several classcal values for cooperatve games can be found n Dressen (1991). The present paper ntroduces some generalzed versons of both the propertes, the null player axom and consstency, and studes dfferent axomatzatons usng them. The results obtaned here have allowed to get new axomatzatons for several classcal values. The frst man result (Theorem 1 n Sect. 3) gves necessary and suffcent condtons for effcent, lnear and symmetrc values to satsfy some generalzed verson of the classcal null player property. Next, a number of ts applcatons to several classcal values for TU-games and ther convex combnatons s shown. Among others, we get an axomatzaton for a new value for TU-games called the per-capta value, aswell as new axomatzatons of the equal splt value and the soldarty value. In Hart and as-colell (1989) and n Sobolev (1973), one can fnd the man results related to the consstency of the Shapley value. Hart and as-colell construct an elegant theory of potental for cooperatve games and apply t to get an axomatzaton of the Shapley value wth the consstency axom. Ther theory s extended n Dressen and Radzk (2003) where the authors gve an axomatzaton based on generalzed consstency propertes for some wde class of values of cooperatve games. The man results gven there have been completed and remarkably strengthened n Sect. 4 of our paper. The group of three results (Propostons 5 and 6, and Theorem 2) establshes necessary and suffcent condtons for the class of effcent, lnear and symmetrc values to satsfy the so-called λ-standardness and a general consstency property. The latter generalzes the consstency property consdered by Hart and as-colell (1989)
odelng values for TU-games usng generalzed versons 181 and s based on some modfcaton of Sobolev s approach to arbtrary values of TUgames. The results of Sect. 4 have been used to get next two results (Theorems 3 and 4 n Sect. 5), that are related to a constructon of a new subclass of values satsfyng the λ-standardness and the so-called probablstc consstency. As an applcaton of ths result, we get new axomatzatons of the Shapley value, the per-capta value, the soldarty value and the equal splt value n terms of these two propertes. Also a wde dscusson of the results obtaned here s gven. Secton 2 s our prelmnary one, where we recall some standard defntons and several basc axoms desred for the values of TU-games, and we quote three results from the lterature, fundamental for our consderatons. We also recall the formulas for fve partcular values, essental for llustratng the results obtaned n the paper. The last Sect. 6 s devoted to the proofs of the theorems. 2 Prelmnares Let N ={1, 2,, n}, wth n 2,beafxedfntesetofn players. Subsets of N are called coaltons whle N s called the grand coalton. The cardnalty of a set X wll be denoted by X. For brevty, throughout the paper, the cardnalty of sets (coaltons) N, S and T wll be also denoted by approprate small letters n, s and t, respectvely. All the set nclusons are meant to be weak. Also, for notatonal convenence, we wll wrte sngleton {} as. A (transferable utlty) game (on N) sanyfunctonv : 2 N R wth v( ) = 0, where R denotes the set of real numbers. Then for any coalton S n N, v(s) descrbes the worth of the coalton S when all the players n S collaborate. A game v s monotonc f v(s) v(t ) for any S T N. Thesetofallgamesv on N s denoted by Ɣ N. For a coalton T N, theunanmty game u T s defned by u T (S) = 1for S T and u T (S) = 0 otherwse, for S N. A value (v) = ( 1 (v),..., n (v)) on Ɣ N s thought of as a vector-valued mappng : Ɣ N R n, whch unquely determnes, for each game v Ɣ N, a dstrbuton of the total wealth avalable to all the players 1, 2,...,n, through ther partcpaton n the game v. We quckly recall several basc propertes a value may have. A value s called effcent f N (v) = v(n) for all games v. If (αv + βw) = α (v) + β (w) for all games v and w and for all reals α and β, avalue s called lnear. If the last equalty holds for α = β = 1, a value s addtve. A player s called a null player (dummy player) ngamev f v(s ) = v(s) (v(s ) = v(s) + v()) for every coalton S N \. If (v) = 0 n case of any null player n game v, we say that a value satsfes the null player axom. If a value satsfes the equalty π (N,πv) = (v) for all N and every permutaton π of the player set N, then we say that satsfes the anonymty axom, sometmes also called the symmetry axom (here πv s defned as game πv by πv(π(s)) = v(s) for S N). If a value satsfes (v) = j (v) f v(s ) = v(s j) for any S N \{, j}, we say that t has the equal treatment property. Ths property s weaker than symmetry.
182 T. Radzk, T. Dressen In ths paper we wll manly dscuss values whch satsfy effcency, symmetry and lnearty. Hence, for brevty, every value satsfyng those three propertes wll be shortly called an ESL-value. Now we recall four classcal ESL-values (the equal splt value, theshapley value, the soldarty value, theδ-dscounted Shapley value) and defne a new one essental for the llustraton of the results obtaned n the paper. The frst two values are standard n cooperatve game theory. The soldarty value was ntroduced n Nowak and Radzk (1994). In the recent paper of Calvo (2008), two varatons of the non-cooperatve model for games n coaltonal form, ntroduced by Hart and as-colell (1996), were proposed, and two new, very nterestng NTU-values have been ntroduced: the random margnal and random removal values. It turned out that for TU-games, the random margnal value concdes wth the Shapley value and that, whch was completely surprsng, the random removal value concdes wth the soldarty value. The fourth value, called the δ-dscounted Shapley value s constructed as a certan modfcaton of the Shapley value, whle the ffth new one, called the per-capta value, s a modfcaton of the soldarty value. The equal splt value Eq. By defnton, t s the ESL-value on Ɣ N defned by Eq (v) = ( Eq (v)) N, where Eq (v) = v(n), N. (1) n So, the equal splt value dvdes the worth v(n) of the grand coalton N equally between all the players, ndependently of the worth of other coaltons. The Shapley value Sh. It s the classcal ESL-value on Ɣ N (ntroduced n Shapley 1953), of the form Sh (v) = ( Sh (v)) N, where Sh (v) = s!(n s 1)! [v(s ) v(s)], N. (2) n! S N\ It s known that the Shapley value Sh on Ɣ N s the unque ESL-value whch satsfes the null player axom. The soldarty value So. Ths s an ESL-value on Ɣ N dscussed n the paper of Nowak and Radzk (1994). It s unquely determned by the three classcal axoms, effcency, addtvty and symmetry, and by a modfcaton of the null player axom, called A-null player axom. We quckly recall ths axom. To express t we need to defne, for any non-empty coalton T N and a game v, the quantty A v (S) = 1 [v(s) v(s \ k)], (3) s k S where s means the cardnalty of S. Clearly, A v (S) can be seen as the average margnal contrbuton of a member of a coalton S. The axom s as follows: A- null player axom: If N s an A-null player n a game v, that s, A v (S) = 0 for every coalton S contanng player, then (v) = 0.
odelng values for TU-games usng generalzed versons 183 It s shown n Nowak and Radzk (1994) that for v Ɣ N, the soldarty value s of the form So (v) = ( So (v)) N, where So (v) = S (n s)!(s 1)! A v (S), N. (4) n! Usng (4), one can easly deduce that the soldarty value has the followng equvalent form So (v) = v(n) n + 1 + s!(n s 1)! v(s ) [ n! s + 2 v(s) ], N. (5) s + 1 S N\ The δ-dscounted Shapley value Shδ.ItsthevalueonƔ N of the form Shδ (v) = (v)) N wth δ R, where ( Shδ Shδ (v) = s!(n s 1)! [δ n s 1 v(s ) δ n s v(s)], N. (6) n! S N\ (Here, by defnton, 0 0 = 1.) An equvalent form of ths generalzaton of the Shapley value was frst ntroduced n Joosten et al. (1994), and next axomatzed (n terms of consstency) by Joosten (1996, Chapter 5). The name δ-dscounted Shapley value comes from Dressen and Radzk (2003). The per-capta value Pc. It s a modfcaton of the soldarty value of the form Pc (v) = ( Pc (v)) N, where Pc (v) = n S N\ s!(n s 1)! n! v(s ) [ s + 1 v(s) ], N. (7) s (Here, by defnton, v(s) s = 0fS =.) Ths new value (not studed n the lterature yet) s dfferent from the per-capta Shapley value ntroduced n Example 3 n Dressen and Radzk (2003). To end wth, we quote three useful facts from the lterature. The frst one belongs to Ruz et al. (1998) (see Lemma 9 there). Proposton 1 A value on Ɣ N s lnear, effcent and satsfes the equal treatment property f and only f there exsts a unque collecton of real constants {λs} s=1,...,n 1 such that for every game v Ɣ N the value payoff vector ( (v)) N s of the followng form: (v) = v(n) λs v(s), N. (8) n n s + S N S λs s v(s) S N =S Remark 1 It s easy to see that any value of the form (8) satsfes the symmetry property. On the other hand, the last property s stronger than equal treatment. Hence Proposton 1 mples the followng: A value on Ɣ N s an ESL-value f and only
184 T. Radzk, T. Dressen f t s effcent, lnear and satsfes the equal treatment property. Therefore, n all the theorems, the phrase ESL-value on Ɣ N can be replaced wth that equvalent weaker form. The second result (see statement () of Theorem 3 n Dressen and Radzk (2003)) gves another characterzaton of the set of ESL-values. Proposton 2 A value on Ɣ N s an ESL-value f and only f there exsts a unque collecton of real constants {b s } s=0,1,...,n wth b n = 1 and b 0 = 0 such that for every game v Ɣ N the value payoff vector ( (v)) N s of the followng form: (v) = S N\ s!(n s 1)! [b s+1 v(s ) b s v(s)], N. (9) n! Remark 2 One can easly see that the formula (9) generalzes the classcal formula (2) for the Shapley value, gvng ths value for the constants b s = 1for1 s n, and gvng the equal splt value for the constants b s = 0 for all 1 s < n. On the other hand, t follows from (5) that (9) wth b s = s+1 1 for 1 s < n, descrbes the formula for the soldarty value. One can also check that f we rewrte formula (8) wth the help of new parameters b 1, b 2,...,b n 1, puttng there λs = b s (n s) for 1 s < n, then (8) concdes wth the formula (9). The thrd result (Theorems 1 and 2 n Radzk and Dressen (2013)) gves two more detaled versons of Proposton 2. To quote t we need to ntroduce the next two very desrable propertes of values. Far treatment: Let, j N and v Ɣ N.Ifv(S ) v(s j) for all S N \ \ j then (v) j (v). onotoncty: Letv be a monotonc game, that s satsfyng v(s) v(t ) whenever S T. Then for each player N, (v) 0. It s worth mentonng that the frst property (far treatment) appears n the lterature under dfferent names, such as desrablty (see, e.g. Peleg and Sudhölter 2003), or local monotonocty (see, e.g. Levnský and Slársky 2004), whlemonotoncty s sometmes called weak monotoncty (see Weber 1988). Besdes, four dfferent natural types of monotoncty are studed n alawsk (2013) and an nterestng dscusson about ther mutual relatons s gven. Proposton 3 Let be an ESL-value on Ɣ N wth ts representaton of the form (9). Then (a) satsfes far treatment f and only f b n = 1 and b s 0 for s = 1, 2,...,n 1. (10) (b) satsfes far treatment and monotoncty f and only f b n = 1 and 0 b s 1 for s = 1, 2,...,n 1. (11)
odelng values for TU-games usng generalzed versons 185 Remark 3 Frst note (by Proposton 2) that all the fve values, Eq, Sh, So, Shδ and Pc defned above, are ESL-values on Ɣ N. A straghtforward comparson of the formulas (1), (2) and (5) wth (9) shows that the constants b s n the representaton formula (9) for the values Eq, Sh, So satsfy nequaltes (11). Therefore these three values satsfy the far treatment and monotoncty propertes. In a smlar way we show that the value Shδ wth 0 δ 1 also satsfes those two propertes. But comparng (7) wth (9), we see that the constants b s n the representaton formula for the per-capta value are of the form b s = n s for s = 1, 2,...,n 1. So, by Proposton 3, the per-capta value satsfes the far treatment property but not the monotoncty property. 3 Theorem on ESL-values wth generalzed null player propertes In ths secton we formulate our frst man result (Theorem 1) where a wde subfamly of ESL-values on Ɣ N s axomatzed wth the help of some theoretcal generalzaton of the null player property. Next we show that ths result mmedately mples axomatzatons (some of them are new) of eght values of TU-games. Let N = {1, 2,, n} wth n 2 be a fxed fnte set. Let α R and let β = (β k ) n k=0 be a sequence of real numbers. We begn wth the followng defnton generalzng the classcal noton of null player n a game. Defnton 1 Player s called a β-null player n a game v Ɣ N f β s+1 v(s ) β s v(s) = 0 for any S N \. Consder now the followng property of a value on Ɣ N. ( β,α)- null player payoff: If player s a β-null player n a game v Ɣ N, then (N,v)= α v(n) n. Now we are ready to formulate the followng characterzaton theorem, the proof of whch wll be gven n Sect. 6. Theorem 1 Let α R and let β be a sequence of real numbers wth β 0 = 0 and β k = 0 for 1 k n. A value on Ɣ N s an ESL-value satsfyng the ( β,α)-null player payoff property f and only f for every game v Ɣ N the value payoff vector (v) = ( (v)) N s of the followng form (v) = α v(n) n + 1 α β n S N\ s!(n s 1)! [β s+1 v(s ) β s v(s)]. (12) n! Now we wll present eght mmedate corollares to Theorem 1. They gve some axomatzatons of the equal splt value, the per-capta value, the δ-dscounted Shapley value, the soldarty value, and convex combnatons of some pars of these values. The frst result repeats the classcal one of Shapley (1953). Namely, takng the sequence β wth β k = 1fork 1, and α = 0 n Theorem 1, we mmedately get the followng corollary. Corollary 1 A value on Ɣ N s an ESL-value satsfyng the null player axom f and only f s the Shapley value Sh ={ Sh } N of the form (2).
186 T. Radzk, T. Dressen Now consder the followng two modfcatons of the null player axom wth a clear nterpretaton. null player average payoff: If player s a null player n a game v Ɣ N, then (v) = v(n) n. null player α- average payoff:let0 α 1. If player s a null player n agamev Ɣ N, then (v) = α v(n) n. Now takng the sequence β wth β k = 1fork 1, and α = 1orα [0, 1] n Theorem 1, we get the next two corollares. Corollary 2 A value on Ɣ N s an ESL-value satsfyng the null player average payoff property f and only f s the equal splt value Eq ={ Eq } N of the form (1). Corollary 3 Let 0 α 1. A value on Ɣ N s an ESL-value satsfyng the null player α-average payoff property f and only f s the α-egaltaran Shapley value α of the form α = α Eq + (1 α) Sh. Remark 4 The null player α-average payoff property can be seen as α-degree of soldarty between the players n a game. So the man dea to consder values wth such property s to add soldarty to the problem of dstrbuton of the grand coalton worth. The α-egaltaran Shapley values α are a consequence of ths approach. The frst axomatzaton of the value α (equvalent to that of Corollary 3) was found n Joosten et al. (1994). It turns out that for any 0 α 1thevalue α has two other, very desred propertes: far treatment and monotoncty. To see ths, t s enough to note that the constants {b s } s=0,1,...,n for α n ts representaton (9) are of the form b n = 1 and b s = 1 α for s < n, and next to apply Proposton 3. Itsalsoworth mentonng that three nterestng axomatzatons of the whole class of the egaltaran Shapley values { α : 0 α 1} have been found recently, two n van den Brnk et al. (2013) and one n Casajus and Huettner (2013). As far as the equal splt value Eq s concerned, ts three other axomatzatons (dfferent from that of Corollary 2) are gven n van den Brnk (2007). Now, let us ntroduce the followng modfcaton of the noton of null player n a game (wth a clear nterpretaton). Defnton 2 Player s called a per-capta null player n a game v Ɣ N f for any S N \, v(s ) s+1 v(s) s = 0. (Here, by defnton, 0 0 = 0.) One can thnk that a per-capta null player s not desrable for any coalton, because after jonng hm the average per player of any coalton s worth does not change, and thereby no coalton may want to jon hm. Consequently, such a player should get nothng. On the other hand, when we assume the hghest degree of soldarty between the players, then a per-capta null player should be awarded v(n) n. The thrd soluton s to award such a player wth somethng between 0 and v(n) n. Ths leads us to the followng three axoms. Per- capta null player: If player s a per-capta null player n a game v Ɣ N, then (v) = 0.
odelng values for TU-games usng generalzed versons 187 Per- capta null player average payoff: If player s a per-capta null player n a game v Ɣ N, then (v) = v(n) n. Per- capta null player α- average payoff: Let0 α 1. If player s a per-capta null player n a game v Ɣ N, then (v) = α v(n) n. Now takng the sequence β wth β k = k 1 for k 1, and α = 0orα = 1or 0 α 1 n Theorem 1, we easly get the next three new axomatzatons. Corollary 4 A value on Ɣ N s an ESL-value satsfyng the per-capta null player property f and only f s the per-capta value Pc ={ Pc } N of the form (7). Corollary 5 A value on Ɣ N s an ESL-value satsfyng the per-capta null player average payoff property f and only f s the equal splt value Eq. By analogy to the α-egaltaran Shapley value, let us call the value = α Eq + (1 α) Pc the α-egaltaran per-capta value. Corollary 6 Let 0 α 1. A value on Ɣ N s an ESL-value satsfyng the percapta null player α-average payoff property f and only f s the α-egaltaran per-capta value α of the form α = α Eq + (1 α) Pc. Remark 5 For 0 α 1, the α-egaltaran per-capta value α satsfes the far treatment property, but rather surprsngly t satsfes monotoncty only for n 1 n α 1, as mpled by Proposton 3. It suffces to note that b s = (1 α) ns for s = 1,...,n 1 n the representaton (9) of α. As a consequence, the per-capta value Pc also satsfes the far treatment property, but not monotoncty. A further modfcaton of the null player noton to that of δ-reducng player, and the correspondng property were ntroduced and dscussed n van den Brnk and Funak (2010). Defnton 3 Let 0 δ 1. Player s called a δ-reducng player n a game v Ɣ N f for any S N \, v(s ) = δ v(s). δ- null player: If player s a δ-reducng player n a game v Ɣ N, then (v) = 0. Now takng the sequence β wth β k = δ k for k 1, and α = 0 n Theorem 1,we mmedately get the result obtaned n van den Brnk and Funak (2010). Corollary 7 Let 0 δ 1. A value on Ɣ N s an ESL-value satsfyng the δ- reducng player property f and only f s the δ-dscounted Shapley value Shδ ={ Shδ } N of the form (6). The last corollary gves a theoretcal characterzaton of the soldarty value. We wll use t n the dscusson of ts structure n Remark 6 below. To formulate t we need to modfy (slghtly) the defnton of per-capta null player n a game and the axom related to t. Defnton 4 Player s called an almost per-capta null player n a game v Ɣ N f for any S N \, v(s ) s+2 v(s) s+1 = 0.
188 T. Radzk, T. Dressen Almost per- capta null player average payoff: If player s an almost per-capta null player n a game v Ɣ N, then (v) = v(n) n+1. Now takng the sequence β wth β k = k+1 1, and α = n new axomatzaton of the soldarty value. n+1 n Theorem 1, we get the Corollary 8 A value on Ɣ N s an ESL-value satsfyng the almost per-capta null player average payoff property f and only f s the soldarty value So ={ So } N of the form (5). Remark 6 The theoretcal axomatzaton of the soldarty value n the last corollary does not seem to have a convncng nterpretaton. However, for large n = N t s very smlar to the axomatzaton of the equal splt value descrbed n Corollary 5. Consderng ths fact, one could thnk that these two values are close for large n. In fact, t turns out that the soldarty value can be seen as asymptotcally equvalent to the equal splt value. Ths problem s studed n Radzk (2013). The asymptotc behavor of some other values, e.g. the dscounted Shapley value, remans an open queston. 4 Consstency theory In the paper of Hart and as-colell (1989) the authors ntroduced some natural noton of consstency of TU-values and showed that the Shapley value s unquely determned by ths consstency property and another one, called the standardness for two-person games. In Dressen and Radzk (2003) we generalzed ther approach, consderng generalzed forms for both of these propertes. The man results obtaned there are completed n ths secton (Proposton 5 and Theorem 2) to f and only f forms. They are formulated n terms of a generalzed consstency and generalzed standardness. In the prevous secton, a value of cooperatve games was understood as a functon defned on the set Ɣ N of all games v wth a fxed grand coalton N ={1, 2,...,n} wth n 2. Now, we extend ths defnton n a natural way. Namely, throughout the next two sectons, a value wll be a functon defned on the class Ɣ of all TU-games (, v) wth fnte grand coaltons {1, 2,...} of cardnalty 2, such that (,v) = ( (,v)) R for each. For any fnte set, letɣ be the setofallgamesoftheform(,v). Now the defnton of an ESL-value on Ɣ must be slghtly modfed. We say that a value satsfes the effcency or the lnearty axom f for every fnte set {1, 2,...}, ts restrcton to Ɣ satsfes one of these axoms, respectvely. A value satsfes the symmetry axom (or s symmetrc) f for all fnte subsets 1 and 2 of the set {1, 2,...} wth 1 = 2, for all games ( 1,v)and 1 1 for every mappng π : 1 2, π ( 2,πv)= ( 1,v), where 1.(Here πv s defned by πv(π(s)) = v(s)). In such cases we wll say that a value s an ESL-value on Ɣ. Now we can repeat Proposton 2 n the verson for a value on Ɣ. Proposton 4 A value on Ɣ s an ESL-value f and only f for any fnte {1, 2,...} there exsts a unque collecton of real constants {b m,s } s=0,1,...,m wth
odelng values for TU-games usng generalzed versons 189 b m,m = 1 and b m,0 = 0 such that for every game (,v) Ɣ, the -th component of the value payoff vector ( (,v)) s of the followng form: (,v)= S \ s!(m s 1)! [b m,s+1 v(s ) b m,s v(s)],. (13) m! We begn wth two general defntons whch are basc for our paper. They come from Dressen and Radzk (2003). The frst one s related to some knd of generalzed standardness for two-person games, and the second one descrbes some general noton of consstency whch extends the reduced game property ntroduced by Hart and as- Colell (1989). Defnton 5 Let λ R. We say that a value s λ-standard for two-person games f for every two-person game ({, j},v)and every player k {, j} t holds that k (v) = λ v(k) + 1 [v(, j) λ v() λ v(j)]. (14) 2 The λ-standardness of a value for two-person games expresses the fact that the value allocates the surplus [v(, j) λ v() λ v(j)] equally to the players and j after each player k concedes to get hs weghted ndvdual worth λ v(k). Remark 7 It s not dffcult to check that the value of the form (13)sλ-standard for two-person games f and only f λ = b 2,1. Therefore the soldarty value s 1 2 -standard, the Shapley value s 1-standard, the per-capta value s 2-standard, and the equal splt value s 0-standard for two-person games. Generally, a value s λ-standard for some λ f and only f s an ESL-value on the set of two-person games. Ths fact can be easly concluded from (14). It s worth mentonng that Joosten et al. (1994) ntroduced some equvalent defnton to λ-standardness of a value for two-person games. Now we ntroduce a property generalzng the consstency property ntroduced by Hart and as-colell (1989). It comes from Dressen and Radzk (2003) and s basc for the rest of our paper. Defnton 6 Let D = {(A k,s, B k,s )} k=2,3,... s=1,...,k be a collecton of pars of real numbers. Gven a game (,v) wth =m 3 and a player, the correspondng reduced game ( \,v, ) assocated wth and D s defned to be as follows: for all nonempty sets S \ v, (S) = A m 1,s [v(s ) (S,v)] + B m 1,s v(s). (15) We say that a value possesses the D-reduced game property on Ɣ or s consstent on Ɣ wth respect to the reduced games v, of the form (15) f t satsfes the followng condton: for every game (,v) Ɣ wth m 3, j ( \,v, ) = j(,v) for all and j \. (16)
190 T. Radzk, T. Dressen (Hart and as-colell (1989) consdered the reduced game v, wth the constants A k,s 1 and B k,s 0.) The reduced games v, of the form (15) have a natural nterpretaton when the constants A k,s, B k,s are nonnegatve and satsfy, A k,s + B k,s = 1 for all k, s. Its presented n the frst paragraph of Sect. 5. Now we are ready to formulate three results about consstency propertes of values, two auxlary ones (Propostons 5 and 6) and Theorem 2 basc for ths secton. Ther proofs are gven n Sect. 6. We need to ntroduce addtonal notaton. Let D be the famly of all collectons D ={(A k,s, B k,s )} k=2,3,... s=1,...,k of pars of real constants. The frst result gves necessary and suffcent condtons for a value to be effcent. It essentally strengthens the result of Lemma 1 n Dressen and Radzk (2003), where only the suffcent condton was dscussed. Notce that λ-standardness of a value (consdered there) s a stronger assumpton than effcency of for two-person games, whch follows from Remark 7. Proposton 5 Let D D and assume that a value possesses D-reduced game property on Ɣ. Then s effcent on Ɣ f and only f s effcent on two-person games and the constants n collecton D satsfy: A k,k = 1 and B k,k = 0 for k = 2, 3,... (17) The second result dscusses the propertes and a possble unqueness of a consstent value. It essentally strengthens and remarkably smplfes the results of Theorems 1 and 2 n Dressen and Radzk (2003), showng that condtons () and () are not necessary to prove the ESL property for a value n Theorem 1, and that only condtons () and () are suffcent to prove the unqueness of n Theorem 2 there. Proposton 6 Let (λ, D) R D wth D fulfllng (17), and assume that a value s λ-standard for two-person games and possesses D-reduced game property on Ɣ. Then the value s unque. oreover t s an ESL-value on Ɣ. In vew of Propostons 5 and 6, one could ask when an ESL-value possesses D-reduced game property for some D. (Note that n vew of Remark 7, t always s λ-standard for two-person games for some λ.) The answer to ths queston s gven n Theorem 2 whch substantally strengthens Theorem 4 n Dressen and Radzk (2003). Theorem 2 Let (λ, D) R D and let be an effcent value on Ɣ. Then the value s λ-standard for two-person games and possesses D-reduced game property on Ɣ f and only s the unque ESL-value on Ɣ such that the constants {b m,s } m=2,3,... s=1,...,m n ts representaton (13), the constant λ and the constants A k,s, B k,s n collecton D satsfy the equaltes (17) and the system of equatons: b 2,1 = λ b m,s A m,s = x m,s (18) b m,s B m,s = b m+1,s x m,s 1 m = 2, 3,... s = 1,...,m 1,
odelng values for TU-games usng generalzed versons 191 where {x m,s } m=2,3,... s=0,1,...,m s the set of real numbers unquely determned by the system { xm,m = m + 1 for m 2 x m,s = (s + 1) m b l+1,s+1 l=s+1 l(l+1) x m,l for m 2 and 0 s m 1. (19) Corollary 9 Let be an ESL-value on Ɣ wth all the constants {b m,s } m=1,2,... s=0,1,...,m n ts representaton (13) dfferent from zero. Then (a) there s a unque par (λ, D) R D wth D fulfllng (17) such that s λ-standard for two-person games and possesses D-reduced game property on Ɣ; (b) the par (λ, D) s the unque soluton of the system (18), (19). Proof Frst note that for every m 2 the system of equatons (19) allows to calculate n turn the constants: x m,m, x m,m 1,...,x m,0. Hence, the constant λ and all the constants A m,s, B m,s are unquely determned by the system of equatons (18), because b m,s = 0 for all m and s. Consequently, by ( ) of Theorem 2, the proof s completed. Remark 8 If some of the constants b m,s n representaton (13) of an ESL-value on Ɣ happen to be zero, the system of equatons (18), (19) may fal. To see ths, t suffces to consder the case of wth b m,1 = b m,2 =... = b m,m 2 = 0 and b m,m 1 = 0 for all m 2. Namely, by the second equalty n system (18), x m,m 2 = 0. But then (19) mplesthatx m,m 2 = b m,m 1 b m+1,m m = 0, a contradcton. Therefore (by Theorem 2) such value s not consstent wth respect to any reduced game of the form (15). On the other hand, t may also happen that every collecton D of the constants A k,s, B k,s s a soluton of the system (18), (19). Namely, consderng = Eq,we have b m,1 = b m,2 =... = b m,m 1 = 0 for all m 2 n ts representaton (13), and t s easy to verfy that, n fact, arbtrary constants A k,s, B k,s satsfy the system (18) then. Remark 9 It s worth mentonng that the formulas for all the constants A k,s, B k,s of collecton D can be obtaned explctly from the system (18), (19) when the parameters b k,s of are of the form b k,s = λk β s = 0 for all k, s (see Corollary 1 n Dressen and Radzk 2003). However, n general, they have a very complex form wthout any natural nterpretaton. The problem can be hghly smplfed when we restrct our consderatons to the natural class consstng of ESL-values on Ɣ wth the addtonal property that the constants b k,s n ther representaton (13) are ndependent of k for all s < k. One can easly see (consderng (1), (2) and (5)) that the Shapley value, the equal splt value and the soldarty value have such a property. It turns out that Theorem 2 can be used to show (n the next secton) that there are very smple necessary and suffcent condtons for a value n ths class to be probablstcally consstent, that s to satsfy A k,s + B k,s = 1 for all k, s. 5 Probablstcally consstent ESL-values In ths secton we wdely dscuss (n two theorems and several corollares and remarks) values for TU-games possessng D-reduced game property wth some collectons of
192 T. Radzk, T. Dressen nonnegatve real numbers n D satsfyng the followng addtonal condton: A k,s + B k,s = 1 for all k, s. In ths case, the nterpretaton for the above consstency property seems to be very natural: Let be a fxed value on a game (,v)and suppose that one of the players, say player, s removed. Further suppose that for every coalton S \ there s stll a chance (wth a probablty equal to A m 1,s ) that wll jon that coalton S or not (wth a probablty equal to B m 1,s ). Snce a new game ( \,v)arses, we can ask, how to estmate the worth of a coalton S n ths new reduced game. In case when player does not jon the coalton S, ts worth s the same as before, that s v(s). However, when player jons the coalton S, theworth of the coalton S after removng that player could be n a natural way estmated as the dfference v(s ) (S,v). Thus the worth of the coalton S n the reduced game wth the player set equal to \ s smply nterpreted as the expected worth gven n the defnton above. Now the consstency property requres that any player j dfferent from should get the same n both games wth respect to the consdered value. In vew of the above comment, we ntroduce the followng defnton of probablstc consstency. Defnton 7 A value s probablstcally consstent on Ɣ f there s a collecton of pars of nonnegatve real numbers D = {(A k,s, B k,s )} k=2,3,... s=1,...,k satsfyng A k,s + B k,s = 1 for all 2 s k, k 2, (20) such that possesses the D-reduced game property on Ɣ wth respect to the reduced games v, of the form (15). Now we are ready to formulate our next two results (Theorems 3 and 4 - ther proofs are shfted to Sect. 6) determnng a set of probablstcally consstent values n some subfamly of the famly of the ESL-values. The frst theorem characterzes the famly of δ-dscounted Shapley value Shδ on Ɣ n terms of the classcal consstency property ntroduced n Hart and as-colell (1989); that s, wth respect to the reduced games v, of the form v, (S) = v(s ) (S,v) for S \. (21) Note that these reduced games are of the form (15) wth A m 1,s 1 and B m 1,s 0, so any consstent value wth respect to such v, s also probablstcally consstent. It s also worth mentonng here that a characterzaton equvalent to the one presented n (a) of Theorem 3, was frst gven (wthout proof) n Joosten et al. (1994), and next n Joosten (1996, Proposton 5.32) wth the proof followng the Hart and as-collel approach. However, t turns out that ths characterzaton s also a smple consequence of Theorem 2. Therefore, n spte of the fact that the part ( ) ofthe statement (a) of Theorem 3 was shown n Example 1 of Dressen and Radzk (2003), for the sake of completeness we wrte t n the followng form:
odelng values for TU-games usng generalzed versons 193 Theorem 3 Let δ R. Then (a) an ESL-value on Ɣ s δ-standard for two-person games, and consstent wth respect to the reduced games v, of the form (21) f and only f s the δ- dscounted Shapley value Shδ of the form (6). (b) the value Shδ satsfes the far treatment and monotoncty propertes f and only f 0 δ 1. The second theorem characterzes some famly of values on Ɣ havng the probablstc consstency property wth respect to generalzed reduced games v, of the form (15) wth arbtrary constants A m 1,s and B m 1,s satsfyng (20). In vew of the comment made n Remark 9, we restrct our consderaton to the famly of ESL-values wth the constants b m,s (n ther representaton (13)) ndependent of m for all s < m. So puttng b m,s = β s n (13), we get the famly of ESL-values on Ɣ descrbed by sequences β = (β s ) s 0 n the followng way: For any game (,v) Ɣ and, the value payoff vector (,v)= ( (,v)) n ths famly s of the followng form: (,v)= v() m (1 β m) + S \ s!(m s 1)! [β s+1 v(s ) β s v(s)]. m! (22) [Note that after replacng b m,s by β s for s < m n the formula (13), t becomes equvalent to (22).] Now we are ready to formulate our next theorem. Theorem 4 Let λ R and assume that a value = ( (v)) on Ɣ s of the form (22) for some real sequence {β n } n 0 wth β 0 = 0. Then () s a λ-standard for two-person games and probablstcally consstent value f and only f β 1 = λ, 0 λ 1 (23) and β s = βs λ := λ for s = 1, 2,.... (24) (1 λ)s + λ () If the equaltes (24) hold for some 0 λ 1, then the value s probablstcally consstent wth respect to the reduced games v,, λ = v, of the form (15) wth the constants A k,k = 1 and B k,k = 0 for k = 2, 3,..., A m 1,s = A λ m 1,s := (1 λ)s + λ (1 λ)(m 1) + λ and B m 1,s = Bm 1,s λ := 1 Aλ m 1,s (25) for m 2 and 1 s m 1. If addtonally λ = 0, v,, λ are the only reduced games wth respect to whch the value gven by (22) (24) s probablstcally consstent.
194 T. Radzk, T. Dressen Now we gve several corollares where, among other thngs, the Shapley value, the per-capta value, the soldarty value and the equal splt value are dscussed n terms of probablstc consstency. Before that some notaton wll be ntroduced. For a real λ and a sequence β λ 1,βλ 2,... determned by (24), let λ be the value defned on Ɣ such that for every game (,v) Ɣ the -th component of the value vector ( λ (T,v)) T s of the followng form: λ v() (T,v)= m [1 βλ m ]+ S T \ Further, let us defne the followng class of values on Ɣ: s!(t s 1)! [βs+1 λ t! v(s ) βλ s v(s)]. (26) F := { λ : 0 λ 1}. (27) The frst two corollares are mmedate consequences of Theorem 4. Corollary 10 A value of the form (22) s probablstcally consstent f and only f F. oreover, every value n F satsfes the far treatment and monotoncty propertes. Corollary 11 For any 0 λ 1 the value λ, determned by (26) and (24), s the only value on Ɣ whch s λ-standard for two person games and probablstcally consstent wth respect to the reduced games v,, λ of the form v,, λ (S) = (1 λ)s+λ [ v(s ) λ (1 λ)(m 1)+λ (S,v) ] + (1 λ)(m 1 s) (1 λ)(m 1) + λ v(s) (28) for S \. If 0 < λ 1, v,, λ are the only reduced games wth respect to whch the value λ s probablstcally consstent. The next corollary shows that the Shapley value, the equal splt value and the soldarty value belong to the class F. Corollary 12 () Sh = 1 ; () Eq = 0 ; () So = 1/2. Proof It s enough to verfy (wth the help of (24)) that the formula (26), wth λ = 1, λ = 0 and λ = 2 1, concdes wth the formulas (2), (1) and (5), respectvely. Remark 10 In vew of (27) and Corollares 10 and 12, t s rather surprsng that such classcal values as the Shapley value and the equal splt value are two endponts of the class F. However, t s much more surprsng that the soldarty value s the central element of F and les exactly n the mddle between the Shapley value and the equal splt value. On the other hand, for 0 <α<1theα-egaltaran Shapley value α = α Eq +(1 α) Sh does not belong to F, and thereby t s not probablstcally consstent. To see ths, note that α s of the form (22) wth β s = 1 α for s > 0, but then the system of equatons (23), (24) has no solutons n λ (because 0 < 1 α<1). In the same way we can check that for 0 α<1, the α-egaltaran per-capta value α = α Eq + (1 α) Pc s not probablstcally consstent ether.
odelng values for TU-games usng generalzed versons 195 The next corollary gves a characterzaton of the equal splt value n terms of λ-standardness and probablstc consstency. Corollary 13 The equal splt value Eq s the only value whch s 0-standard for two-person games and probablstcally consstent wth respect to the reduced games of the form v, = veq, v Eq, (S) = v(s ) Eq (S,v) for S \. (29) Proof It s an mmedate consequence of Theorem 3 wth δ = 0. Remark 11 Obvously, the equal splt value Eq s 0-standard for two-person games. It s nterestng that for ths value Corollary 11 gves also (after puttng λ = 0n(28)) other reduced games v, of the form v Eq,,0 (S) = s [ m 1 = veq,,0 v(s ) Eq ] (S,v) + m s 1 m 1 v(s). So reduced game for Eq s not unque. Nevertheless ths does not contradct Theorem 4 because λ = 0for Eq. oreover, t s shown n Remark 8 that for the value Eq, arbtrary constants A m,s and B m,s satsfy the system of equatons (18). Therefore, by Theorem 2, the equal splt value s probablstcally consstent wth respect to any reduced games of the form (15) wth arbtrary constants 0 A m,s 1 and B m,s = 1 A m,s satsfyng (17). The next two corollares gve characterzatons of the Shapley value and the soldarty value. These two results were obtaned n Dressen and Radzk (2003) (see Examples 1 and 2 there) wthout the unqueness n ther formulatons. Corollary 14 The Shapley value Sh s the only value whch s 1-standard for twoperson games, and probablstcally consstent wth respect to the reduced games v Sh, of the form v Sh, (S) = v(s ) Sh (S,v) for S \. (30) The games v Sh, are the only reduced games wth respect to whch the value Sh s probablstcally consstent. Proof One can easly check that the reduced games v,, λ of the form (28) wth λ = 1 concde wth the ones of the form (30). Hence the statement () of Corollary 12 and Corollary 11 mmedately complete the proof. Remark 12 The result of Corollary 14 n a slghtly weaker form was earler proved n Hart and as-colell (1989). Namely, the unqueness of the Shapley value was shown there n the class of values whch are relatvely nvarant under strategc equvalence and have the equal treatment property nstead of 1-standardness. Ther result s a smple consequence of a very nce theory of potental for cooperatve games, constructed n
196 T. Radzk, T. Dressen that paper. It s also worth mentonng here that two other extensons of reduced games (30) n terms of consstency were dscussed n the lterature. Namely Joosten et al. (1994) consdered some α-reduced games, whle Derks and Peters (1997) appled the consstency property to TU-games wth restrcted coaltons. The next corollary strengthens the results presented n Example 2 n Dressen and Radzk (2003). Corollary 15 The soldarty value So s the only value whch s 2 1 -standard for two-person games and probablstcally consstent wth respect to the reduced games v, = vso, of the form v So, (S) = s + 1 [ m v(s ) So The games v So, s probablstcally consstent. ] (S,v) + m s 1 v(s) for S \. m (31) are the only reduced games wth respect to whch the value So Proof It s easy to verfy that the reduced games v,, λ of the form (28) wth λ = 2 1 concde wth the ones of the form (31). Hence the statement () of Corollary 12 and Corollary 11 mmedately complete the proof. Remark 13 It s worth mentonng that Sobolev (1973) consdered a slghtly dfferent consstency property, and showed that the Shapley value s the only one whch s 1- standard for two person games and consstent wth respect to the reduced games of the form v, (S) = s [ m 1 v(s ) Sh surprsngly smlar to the reduced games v So, value, and to the reduced game v Eq, ] (,v) + m s 1 m 1 v(s), of the form (31), related to the soldarty,0 (from Remark 11), mpled by Theorem (4) for the equal splt value. Sobolev s result was recently generalzed by van den Brnk et al. (2013) who showed that for any 0 α 1, the α-egaltaran Shapley value α = α Eq + (1 α) Sh s the unque value whch s α-standard for two person games and consstent wth respect to the same reduced games v,.itsalsoworth mentonng that Joosten (1996) showed that for 0 δ 1, δ-standardness combned wth Hart and as-colell consstency (wth respect to reduced games of the form (30)) characterze the δ-dscounted Shapley value Shδ of the form (6). 6 Proofs of the theorems In ths secton we gve the proofs of Theorems 1 4 formulated n Sects. 3 5. Proof of Theorem 1 ( ) Let N ={1, 2,, n} wth n 2, and let = ( (v)) N be a value on Ɣ N of the form (12). We can easly check that concdes wth the value of the form (9) wth b n = 1 and wth b s = 1 α β n β s for 1 s < n. Therefore, by
odelng values for TU-games usng generalzed versons 197 Proposton 2, s an ESL-value on Ɣ N. The fact that also satsfes the ( β,α)-null player axom follows mmedately from (12). ( ) Assume now that s an ESL-value on Ɣ N satsfyng the ( β,α)-null player property. By Proposton 2, there exsts a unque collecton of real constants {b s } s=0,1,...,n wth b n = 1 and b 0 = 0 such that for all N, (v) s of the form (9). But then (v) can be easly rewrtten n the form (v) = v(n) n (1 b n ) + S N\ s!(n s 1)! [b s+1 n! v(s ) b sv(s)] (32) for all N and v Ɣ N, where b s = b s for s = 0, 1,...,n 1 and b n s arbtrally chosen. Therefore we can take b n = 1 + α,. Let us defne another value = ( (v)) N on Ɣ N as n (12), that s by v(n) (v) = α + 1 α n β n S N\ s!(n s 1)! [β s+1 v(s ) β s v(s)] (33) n! for all N and v Ɣ N. It s clear that = f b s = 1 α β n β s for s = 1, 2,...,n 1. Ths s thus what we show n the remander of the proof. Let us denote α n = 0, α 0 = 0 and α s = b s 1 α β n β s for s = 1,...,n 1. (34) Now consder the vector functon E(v) ={E (v)} N on Ɣ N defned by E (v) = S N\ s!(n s 1)! [α s+1 v(s ) α s v(s)], N. (35) n! One can easly check that E(v) = (v) (v) for v Ɣ N. Hence, we need only to show that α s = 0 for s = 1,...,n 1. (36) By assumpton, s an ESL-value on Ɣ N and has the ( β,α)-null player property. By the part ( ),thevalue has the same propertes. Consequently, for every game v Ɣ N E (v) = 0 f s a β -null player n v. (37) Let us consder the class of games V ={v K : = K N}, defned on N by the followng: v K (S) = β 1 s f K S N and v K (S) = 0, otherwse. Let K s = N \{1, 2,...,s} for s = 1,...,n 1. One can easly state that player 1sa β-null player n all the games v K1,v K2,,...,v Kn 1. Therefore by (37) E 1 (v Ks ) = 0 for s = 1,...,n 1. (38)
198 T. Radzk, T. Dressen But (35) leads to the equalty E 1 (v K1 ) = 1 n whence, by (34) and (38) wth s = 1, we get Further, (35) mples the followng: E 1 (v K2 ) = 1 n [ αn α ] n 1, (39) β n β n 1 α n 1 = 0. (40) [ αn α ] [ n 1 2 αn 1 + α ] n 2. (41) β n β n 1 n(n 1) β n 1 β n 2 But ths, n vew of (34), (40) and (38) wth s = 2, mmedately gves α n 2 = 0. (42) We can contnue these calculatons successvely for E 1 (v K3 ),...,E 1 (v Kn 1 ) to get α n s = 0 for s = 3,...,n 1. (43) Therefore we have proved (36) and thereby the proof of the theorem s completed. Proof of Proposton 5 ( ) Let be an effcent value satsfyng the assumpton. We need only to show that (17) holds. Let us fx any fnte set wth =m 3 and a game v Ɣ. Choose and consder the reduced game v, on the set \ of the form (15). By the assumpton, s effcent on the class of games Ɣ \. Therefore, by (15) and (16), j (,v) = j ( \,v, ) = v, ( \ ) j \ j \ = A m 1,m 1 [v() (,v)] + B m 1,m 1 v( \ ). for. Hence, usng the effcency of, we get the followng sequence of equaltes: (m 1)v() = mv() (,v)= [v() (,v)] = j (,v) (,v) = j (,v) j j \ = [ Am 1,m 1 [v() (,v)] + B m 1,m 1 v( \ ) ]
odelng values for TU-games usng generalzed versons 199 Therefore [ = A m 1,m 1 mv() ] (,v) + B m 1,m 1 v( \ ) = A m 1,m 1 (m 1)v() + B m 1,m 1 v( \ ). (m 1)(1 A m 1,m 1 )v() = B m 1,m 1 v( \ ). Hence, snce m 3 and the game (,v)s arbtrarly fxed, we easly deduce that both sdes of the last equalty must be equal to 0. Consequently, (17) holds because of the arbtrarty of m. ( ) To prove ths part we can repeat the proof of Lemma 1 n Dressen and Radzk (2003), wth the excepton that the λ-standardness for should be changed by effcency of for two person games. Ths ends the proof of Proposton 5. Proof of Proposton 6 Let be a value satsfyng the assumpton. It s easly seen from (14) that s effcent for two person games. Hence, n vew of the part ( ) of Proposton 5, s effcent on Ɣ, Therefore, to prove the theorem t suffces only to show for every fnte {1, 2,...} wth 2, that s unquely determned, lnear and symmetrc on Ɣ. We wll show t by nducton wth respect to the number m =, m = 2, 3,... When m = 2, s unquely defned on every Ɣ wth =2, because of the λ-standardness of on the set of all two-person games, and t s obvously lnear and symmetrc on Ɣ (see (14) n Defnton 5). Now, by the nducton hypothess, assume that for some m 3, the value s unquely determned, lnear and symmetrc on every set Ɣ T wth T < m. By Proposton 1, there s a unque collecton of constants {λ t s t = 2, 3,...,m 1, s = 1, 2,...,t 1} such that for any T < m and v Ɣ T (T,v)= v(t ) t + S T S λ t s v v(s) λ t s v v(s), T. (44) s t s S T =S Now, let be any fnte set wth = m. We wll show that our nductve assumpton mples unqueness, as well as lnearty and symmetry of on Ɣ. Let us arbtrarly fx a game v Ɣ, and denote j (,v)= x j, j. (45) Choose and consder the reduced game v, on the set \ of the form (15). By the nductve assumpton, the value s unquely determned on the class of games Ɣ \, and s an ESL-value on ths set. Hence, by (44),
200 T. Radzk, T. Dressen and j ( \,v, ) = 1 m 1 v, ( \ ) + S \ S j λ m 1 s S \ S j λs m 1 v, s (S) m 1 s v, (S). (46) By the effcency of,wehave (,v)= v(). (47) N Further, by (16), (17) and (15), we have: j ( \,v, ) = j(,v) for j \, (48) v, ( \ ) = v() (,v) (49) v, (S) = A m 1,s [v(s ) (S,v)] + B m 1,s v(s) for S \. (50) Besdes (44) mplesthat (S,v)= 1 v(s ) + s + 1 U S U λu s+1 u v(u) U S U λu s+1 v(u) (51) s + 1 u for all S \. Now, puttng (51) n(50), and next, puttng (48), (49) and (50) n(46), we get j (,v)+ 1 m 1 (,v)= a j (v) (52) for and j \, where a j (v) s the unquely determned lnear functon of v of the form a j (v) = S S, S j β s v(s) + S S, S j γ s v(s) + S S, S j δ s v(s) + S S, S j τ s v(s) (53) wth some constants β s, γ s, δ s and τ s, dependent only on the cardnalty s = S of sets S. Hence, after usng (45), equaltes (52) lead to the system of m(m 1) lnear equatons wth m varables (x ) of the followng form: 1 m 1 x + x j = a j (v), j \. (54)